Tagged Questions

Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

Claim: The empty set is open.
Proof. Assume that the empty set is closed. Then, there must be one point such that any point in its ball is not inside of the empty set. However, the empty set has no ...

Question:Prove that if $\vec{g} : \mathbb{R}^n \rightarrow \mathbb{R}^n $ and $ \det(\vec{g}') \neq 0$, then in some open set $V \subset \mathbb{R}^n $ such that $\vec{x} \in V$ we have:
$\vec{g} = ...

Given the function of a double cone and a plane, how do we find the intersection between them?
Suppose the equation of the cone is $f(x, y, z) = 0$ and the equation of the plane is $h(x, y, z) = 0$. ...

I'm working through some practice problems in one of my math textbooks, and I'm told to find both the parametric and symmetric equations of the line through $(1,-1,1)$ and parallel to the line $ x + 2 ...

plane $x+y-z=-2$ intersects $z^2=x^2+y^2$
I need to use Lagrange multipliers to determine the point of intersection which is the closest to the origin.
As far as I understand, to use Lagrange I need ...

There is a theorem in our textbook saying that rather than calculating all the derivatives needed to compute the taylor polynomial, if one can find, by any means, a polynomial $Q$ of degree $k$ such ...

If the equation $F(x,y,z)=0$ defines $z$ implicitly as a differentiable function of x and y, then by taking a partial derivative with respect to one of the independent variables (in this case x), you ...

this is my first post. I have been working through Spivak's Calculus on Manifolds and have finally arrived at a section devoted to partitions of unity. Up until now, I have not had very much trouble ...

The sphere given by $x^{2} + y^{2} + z^{2} = 4$ is submerged in an electric field with charge density given by $f(x, y, z) = x^{2} + y^{2}$. Find the total amount of electric charge on this surface.
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